Second order linear q-difference equations: nonoscillation and asymptotics
| Authors | |
|---|---|
| Year of publication | 2011 |
| Type | Article in Periodical |
| Magazine / Source | Czechoslovak Mathematical Journal |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | regularly varying functions; $q$-difference equations; asymptotic behavior; oscillation |
| Description | This paper can be understood as a completion of $q$-Karamata theory along with a related discussion on asymptotic behavior of solutions to the linear $q$-difference equations. The $q$-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $\qN:=\{q^k:k\in \N_0\}$ with $q>1$. In addition to recalling the existing concepts of $q$-regular variation and $q$-rapid variation we introduce $q$-regularly bounded functions and prove many related properties. The $q$-Karamata theory is then applied to describe (in an exhaustive way) the asymptotic behavior as $t\to\infty$ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p:\qN\to\R$. We also present existing and new criteria of Kneser type which are related to our subject. A comparison of our results with their continuous counterparts is made. It reveals interesting differences between the continuous case and the $q$-case and validates the fact that $q$-calculus is a natural setting for Karamata like theory and provides a powerful tool in qualitative theory of dynamic equations. |
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